Chapter 7: Ecosystems

7.2.1 Constant production: linear functional response

One-level system

This system w single functional group, “plants.” Carbon biomass as currency. Carbon biomass density as \(P(t)\) gC/m2. Photosynthesis produces new biomass at \(\phi\) gC/m2/day, and plant of mass \(w\) loses mass (mortality and respiration) at \(\delta_p w\) gC/day.

\[\frac{dP}{dt} = \phi - \delta_p P\] So, steady state stock of plants at \[0 = \phi - \delta_p P \Longrightarrow P^* = \phi / \delta_p\]

Two-level system

Add a second group, “herbivores,” to the ecosystem. Carbon biomass density of herbivores is \(H(t)\) gC/m2; respiration & mortality loss at \(\delta_h\) day-1, and linear functional response attack rate \(\alpha_h\) m^2/day/gC.

\[\frac{dP}{dt} = \phi - \delta_p P - \alpha_h PH\] and \[\frac{dH}{dt} = \alpha_h PH - \delta_h H\] Can find two steady states, one with \(H=0\) and \(P\) given by one-level steady state above. The other is: \[P^* = \delta_h / \alpha_h, \quad H^* = \frac{1}{\delta_h}(\phi - \delta_p P^*)\]

Stability analysis using Jacobian (from Primer)

This is not in Ecological Dynamics, but based on Primer.

From the equil formulas above, \(P^* = \frac{\delta_h}{\alpha_h}\) = 50, \(H^* = \frac{1}{\delta_h}(\phi - \delta_p P^*)\) = 10

Creating, solving and using the Jacobian matrix

To know how each population growth rate changes in response to changes in the abundance each of the other population, we take the partial derivatives.

\[\mathbf J = \begin{pmatrix} \frac{\partial \dot P}{\partial P} & \frac{\partial \dot P}{\partial H} \\ \frac{\partial \dot H}{\partial P} & \frac{\partial \dot H}{\partial H} \end{pmatrix} = \begin{pmatrix} -\delta_p - \alpha_h H & -\alpha_h P \\ \alpha_h H & \alpha_h P - \delta_h \end{pmatrix}\]

We can replace the \(P\) and \(H\) in the Jacobian with the equilibria found above \(P^*, H^*\).

\[\begin{pmatrix} -\delta_p - \alpha_h \frac{1}{\delta_h}(\phi - \delta_p \frac{\delta_h}{\alpha_h}) & -\alpha_h \frac{\delta_h}{\alpha_h} \\ \alpha_h \frac{1}{\delta_h}(\phi - \delta_p \frac{\delta_h}{\alpha_h}) & \alpha_h \frac{\delta_h}{\alpha_h} - \delta_h \end{pmatrix} = \begin{pmatrix} \frac{-\alpha_h \phi}{\delta_h} & -\delta_h\\ \frac{\alpha_h \phi}{\delta_h} - \delta_p & 0 \end{pmatrix}\]

Recall the Routh-Hurwitz criterion:

\[ J_{11} + J_{22} < 0\] \(J_{11}\) is negative, so the plant population will self regulate, while \(J_{22}\) is zero, so the herbivore population does not have a density-dependence effect on itself. ???

The other part of the Routh-Hurwitz criterion is the condition:

\[ J_{11}J_{22} - J_{12}J_{21} > 0\] In the plant-herbivore context, this suggests that the plant declines due to the herbivore (\(J_{12} < 0\)) and the herbivore increases due to the plant (\(J_{21} > 0\) as long as \(\phi > \delta_p \frac{\delta_h}{\alpha_h}\)). The signs of these elements make their product negative, and help make the above condition true.

Note: the \(\phi > \frac{\delta_p \delta_h}{\alpha_h}\) term indicates: primary production \(\phi\) is high enough to support respiration/mortality losses from the plant population at the density needed to bring the herbivore production and loss into balance, or alternately, there’s only a finite herbivore standing stock if the plant standing stock in the absennce of herbivory is larger than that required to sustain steady-state herbivore standing stock.

If we performed eigenanalysis on the above Jacobian matrix, we would find that the eigenvalues are complex (-0.1+0.2i, -0.1-0.2i), meaning that the populations will oscillate or cycle, with period \(\frac{2\pi}{\omega}\). * If the real parts are zero, the system will exhibit neutral stability * If the real parts are negative, the system will converge toward the equilibrium. * If the real parts are positive, the system will approach a stable oscillation away from the equilibrium.

Three-level system

Add a third group, “consumers,” which only eat herbivores, to the ecosystem. Carbon biomass density of consumers is \(C(t)\) gC/m2; respiration & mortality loss at \(\delta_c\) day-1, and linear functional response attack rate \(\alpha_c\) m^2/day/gC. The system of equations:

\[\frac{dP}{dt} = \phi - \delta_p P - \alpha_h PH\] \[\frac{dH}{dt} = \alpha_h PH - \delta_h H - \alpha_c H C\] \[\frac{dC}{dt} = \alpha_c HC - \delta_c C\] Can find three steady states:

  • Plants only, with \(C = H = 0\) and \(P\) given by one-level steady state above.
  • No consumers, with \(C = 0\) and the system devolves to a two-level system
  • Coexistence steady state, with \[P^* = \frac{\phi}{\delta_p + \alpha_h H^*}, \quad H^* = \delta_c / \alpha_c, \quad C^* = \frac{1}{\delta_c}(\phi - \delta_p P^* - \delta_h H^*)\]

Three-level dynamics

To have sensible equilibria, the following needs to hold true: \[\phi \geq \delta_p \left(\frac{\delta_h}{\alpha_h}\right) + \delta_h \left(\frac{\delta_c}{\alpha_c} \right)\]

Stability analysis using Jacobian

From the equil formulas above:

  • \(P^* = \frac{\phi}{\delta_p + \alpha_h H^*}\) = 294.1176471
  • \(H^* = \frac{\delta_c}{\alpha_c}\) = 62
  • \(C^* = \frac{1}{\delta_c}(\phi - \delta_p P^* - \delta_h H^*)\) = 38.0392157.
Creating, solving and using the Jacobian matrix

To know how each population growth rate changes in response to changes in the abundance each of the other population, we take the partial derivatives.

\[\mathbf J = \begin{pmatrix} \frac{\partial \dot P}{\partial P} & \frac{\partial \dot P}{\partial H} & \frac{\partial \dot P}{\partial C} \\ \frac{\partial \dot H}{\partial P} & \frac{\partial \dot H}{\partial H} & \frac{\partial \dot H}{\partial C} \\ \frac{\partial \dot C}{\partial P} & \frac{\partial \dot C}{\partial H} & \frac{\partial \dot C}{\partial C} \end{pmatrix} = \begin{pmatrix} -\delta_p - \alpha_h H & -\alpha_h P & 0\\ \alpha_h H & \alpha_h P - \delta_h - \alpha_c C & -\alpha_c H\\ 0 & \alpha_c C & \alpha_c H - \delta_c \end{pmatrix}\]

We can replace the \(P, H, C\) in the Jacobian with the equilibria found above \(P^*, H^*, C^*\).

\[\mathbf J = \begin{pmatrix} -\delta_p - \frac{\alpha_h \delta_c}{\alpha_c} & - \frac{\alpha_h \phi}{\delta_p + \frac{\alpha_h \delta_c}{\alpha_c}} & 0\\ \frac{\alpha_h \delta_c}{\alpha_c} & \frac{\alpha_h \phi}{\delta_p + \alpha_h \frac{\delta_c}{\alpha_c}} - \delta_h - \frac{\alpha_c }{\delta_c}(\phi - \frac{\delta_p \phi}{\delta_p + \frac{\alpha_h \delta_c}{\alpha_c}} - \frac{\delta_h\delta_c}{\alpha_c}) & -\frac{\alpha_c \delta_c}{\alpha_c}\\ 0 & \frac{\alpha_c}{\delta_c}(\phi - \frac{\delta_p \phi}{\delta_p + \frac{\alpha_h \delta_c}{\alpha_c}} - \frac{\delta_h \delta_c}{\alpha_c}) & \frac{\alpha_c \delta_c}{\alpha_c} - \delta_c \end{pmatrix}\] \[\Longrightarrow \mathbf J = \begin{pmatrix} -\delta_p - \frac{\alpha_h\delta_c}{\alpha_c} & - \frac{\alpha_h \phi}{\delta_p + \frac{\alpha_h \delta_c}{\alpha_c}} & 0\\ \frac{\alpha_h \delta_c}{\alpha_c} & 0 & -\delta_c\\ 0 & \frac{\alpha_h \phi}{\delta_p + \frac{\alpha_h \delta_c}{\alpha_c}} - \delta_h & 0 \end{pmatrix}\]

Recall the Routh-Hurwitz criterion, assuming we can update simply for three species?:

\[ J_{11} + J_{22} + J_{33}< 0\] \(J_{11}\) is negative, so the plant population will self regulate, while \(J_{22}\) and \(J_{33}\) are zero, so the herbivore and consumer populations do not have a density-dependence effect on themselves. ???

The other part of the Routh-Hurwitz criterion is the condition (how to update for three species?):

\[ J_{11}J_{22} - J_{12}J_{21} > 0\]

When we performed eigenanalysis on the above Jacobian matrix, we find that the eigenvalues are complex (0.644+0i, -0.577+0.219i, -0.577-0.219i), meaning that the populations will oscillate or cycle. What does it mean that one eigenvalue is not complex? * If the real parts are zero, the system will exhibit neutral stability * If the real parts are negative, the system will converge toward the equilibrium. * If the real parts are positive, the system will approach a stable oscillation away from the equilibrium.

Four-level system

Add a fourth group, “top predators,” which only eat consumers, to the ecosystem. Carbon biomass density of top predators is \(T(t)\) gC/m2; respiration & mortality loss at \(\delta_t\) day-1, and linear functional response attack rate \(\alpha_t\) m^2/day/gC. The system of equations:

\[\frac{dP}{dt} = \phi - \delta_p P - \alpha_h PH\] \[\frac{dH}{dt} = \alpha_h PH - \delta_h H - \alpha_c H C\] \[\frac{dC}{dt} = \alpha_c HC - \delta_c C - \alpha_c CT\] \[\frac{dT}{dt} = \alpha_t CT - \delta_t T\] Can find four steady states:

  • Plants only, with \(T = C = H = 0\) and \(P\) given by one-level steady state above.
  • No consumers, with \(T = C = 0\) and the system devolves to a two-level system
  • Steady state with no top predators, as in three-level system.
  • Coexistence of all four levels - see Table 7.1.

Four-level dynamics

To have sensible equilibria, the following needs to hold true: ??? update \[\phi \geq \delta_p \left(\frac{\delta_h}{\alpha_h}\right) + \delta_h \left(\frac{\delta_c}{\alpha_c} \right)\]

7.2.2 Logistic primary production

One-level system

Modify plant carbon biomass to grow logistically to carrying capacity \(K\):

\[\frac{dP}{dt} = rP \left( 1 - \frac{P}{K} \right)\]

Two-level system

Add a second group, “herbivores,” to the ecosystem, following assumptions in 7.2.1.

\[\frac{dP}{dt} = rP \left( 1 - \frac{P}{K} \right) - \alpha_h P H\] and \[\frac{dH}{dt} = \alpha_h PH - \delta_h H\] Can find two steady states, one with \(H=0\) and \(P = K\). The other is: \[P^* = \delta_h / \alpha_h, \quad H^* = \frac{r}{\alpha_h} \left(1 - \frac{P^*}{K} \right)\]

Stability analysis using Jacobian (from Primer)

From the equil formulas above, \(P^* = \frac{\delta_h}{\alpha_h}\) = 50, \(H^* = \frac{r}{\alpha_h} \left(1 - \frac{P^*}{K} \right\) = 37.5

Creating, solving and using the Jacobian matrix

To know how each population growth rate changes in response to changes in the abundance each of the other population, we take the partial derivatives.

\[\mathbf J = \begin{pmatrix} \frac{\partial \dot P}{\partial P} & \frac{\partial \dot P}{\partial H} \\ \frac{\partial \dot H}{\partial P} & \frac{\partial \dot H}{\partial H} \end{pmatrix} = \begin{pmatrix} r \left(1 - \frac{2P}{K} \right) - \alpha_h H & -\alpha_h P \\ \alpha_h H & \alpha_h P - \delta_h \end{pmatrix}\]

We can replace the \(P\) and \(H\) in the Jacobian with the equilibria found above \(P^*, H^*\).

\[\begin{pmatrix} r \left(1 - \frac{2\frac{\delta_h}{\alpha_h}}{K} \right) - \alpha_h \frac{r}{\alpha_h} \left(1 - \frac{\frac{\delta_h}{\alpha_h}}{K} \right) & -\alpha_h \frac{\delta_h}{\alpha_h} \\ \alpha_h \frac{r}{\alpha_h} \left(1 - \frac{\frac{\delta_h}{\alpha_h}}{K} \right) & \alpha_h \frac{\delta_h}{\alpha_h} - \delta_h \end{pmatrix} = \begin{pmatrix} \frac{-r \delta_h}{\alpha_h K} & -\delta_h\\ r \left(1 - \frac{\delta_h}{\alpha_h K} \right) & 0 \end{pmatrix}\]

Recall the Routh-Hurwitz criterion:

\[ J_{11} + J_{22} < 0\] \(J_{11}\) is negative, so the plant population will self regulate, while \(J_{22}\) is zero, so the herbivore population does not have a direct density-dependence effect on itself.

\[ J_{11}J_{22} - J_{12}J_{21} > 0\] \(J_{11}J_{22} = 0\), so stability depends on \(J_{12}J_{21} < 0\). The plant declines due to the herbivore (\(J_{12} < 0\)) and the herbivore increases due to the plant (\(J_{21} > 0\) as long as \(P^* < K\)).

\[\begin{pmatrix} \frac{-r \delta_h}{\alpha_h K} & -\delta_h\\ r \left(1 - \frac{\delta_h}{\alpha_h K} \right) & 0 \end{pmatrix}\]

If we performed eigenanalysis on the above Jacobian matrix, we would find that the eigenvalues are complex (-0.062+0.428i, -0.062-0.428i), meaning that the populations will oscillate or cycle, with period \(\frac{2\pi}{\omega}\). * If the real parts are zero, the system will exhibit neutral stability * If the real parts are negative, the system will converge toward the equilibrium. * If the real parts are positive, the system will approach a stable oscillation away from the equilibrium.

Three-level system

Add a third group, “consumer,” which only eat herbivores, to the ecosystem. Carbon biomass density of consumers is \(C(t)\) gC/m2; respiration & mortality loss at \(\delta_c\) day-1, and linear functional response attack rate \(\alpha_c\) m^2/day/gC. The system of equations:

\[\frac{dP}{dt} = rP \left( 1 - \frac{P}{K} \right) - \alpha_h PH\] \[\frac{dH}{dt} = \alpha_h PH - \delta_h H - \alpha_c H C\] \[\frac{dC}{dt} = \alpha_c HC - \delta_c C\] Can find three steady states:

  • Plants only, with \(C = H = 0\) and \(P = K\).
  • No consumers, with \(C = 0\) and the system devolves to a two-level system
  • Coexistence steady state, with
    • \(P^* = K \left(1 - \frac{\alpha_h H^*}{r} \right)\)
    • \(H^* = \delta_c / \alpha_c\)
    • \(C^* = \frac{1}{\delta_c} \left[ rP^* \left( 1 - \frac{P^*}{K} \right) - \delta_h H^* \right]\)

Three-level dynamics

To have sensible equilibria, the following needs to hold true:

\[K \left(1 - \frac{\alpha_h \delta_c}{\alpha_c r} \right) \geq \frac{\delta_h}{\alpha_h}\]

Four-level system (not correct?)

Add a fourth group, “top predators,” which only eat consumers, to the ecosystem. Carbon biomass density of top predators is \(T(t)\) gC/m2; respiration & mortality loss at \(\delta_t\) day-1, and linear functional response attack rate \(\alpha_t\) m^2/day/gC. The system of equations:

\[\frac{dP}{dt} = rP \left( 1 - \frac{P}{K} \right) - \alpha_h PH\] \[\frac{dH}{dt} = \alpha_h PH - \delta_h H - \alpha_c H C\] \[\frac{dC}{dt} = \alpha_c HC - \delta_c C - \alpha_c CT\] \[\frac{dT}{dt} = \alpha_t CT - \delta_t T\] Can find four steady states:

  • Plants only, with \(T = C = H = 0\) and \(P\) given by one-level steady state above.
  • No consumers, with \(T = C = 0\) and the system devolves to a two-level system
  • Steady state with no top predators, as in three-level system.
  • Coexistence of all four levels:
    • \(P^* = \frac{1}{\alpha_h}(\delta_h + \alpha_c C^*)\)
    • \(H^* = \frac{r}{\alpha_h}\left(1 - \frac{P}{K} \right)\)
    • \(C^* = \frac{\delta_t}{\alpha_t}\)
    • \(T^* = \frac{1}{\alpha_t}(\alpha_c H^* - \delta_c)\)

Four-level dynamics??? not correct

To have sensible equilibria, the following needs to hold true: ??? update \[\phi \geq \delta_p \left(\frac{\delta_h}{\alpha_h}\right) + \delta_h \left(\frac{\delta_c}{\alpha_c} \right)\]

7.2.3 Type II functional response

One-level system

Plant carbon biomass as constant primary production (same as 7.2.1)

\[\frac{dP}{dt} = \phi - \delta_p P\]

Two-level system

Add a second group, “herbivores,” to the ecosystem, but now the herbivores have a type II functional response.

\[\frac{dP}{dt} = \phi - \delta_p P - \frac{\alpha_h H P}{P + P_0}\] and \[\frac{dH}{dt} = \frac{\alpha_h PH}{P + P_0} - \delta_h H\] Can find two steady states, one with \(H=0\) and \(P = \phi/\delta_p\). The other is:

\[P^* = \frac{\delta_h P_0}{\alpha_h - \delta_h}, \quad H^* = \frac{\phi - \delta_p P^*}{\delta_h}\]

These should be positive, as long as:

\[\frac{\phi}{\delta_p} \geq \frac{\delta_h P_0}{\alpha_h - \delta_h}\]

Three-level system

The “consumer” functional group now has a type II functional response as well.

  • \(\frac{dP}{dt} = \phi - \delta_p P - \frac{\alpha_h H P}{P + P_0}\)
  • \(\frac{dH}{dt} = \frac{\alpha_h PH}{P + P_0} - \delta_h H - \frac{\alpha_c C H}{H + H_0}\)
  • \(\frac{dC}{dt} = \frac{\alpha_c C H}{H + H_0} - \delta_c C\)

As usual, can find three steady states; coexistence steady state, with

  • \(P^* = K \left(1 - \frac{\alpha_h H^*}{r} \right)\)
  • \(H^* = \frac{\delta_c H_0}{\alpha_c - \delta_c}\)
  • \(C^* = \frac{1}{\delta_c}(\phi - \delta_p P^* - \delta_h H^*)\)

Three-level dynamics

To have sensible equilibria, the following needs to hold true:

\[K \left(1 - \frac{\alpha_h \delta_c}{\alpha_c r} \right) \geq \frac{\delta_h}{\alpha_h}\]